Signed q-Analogs of Tornheim's Double Series

TL;DR

This paper introduces signed q-analogs of Tornheim's double series, evaluates them using double q-Euler sums, and provides explicit evaluations, correcting previous literature errors.

Contribution

It presents the first signed q-analogs of Tornheim's series and offers explicit evaluations, advancing the understanding of q-analogs in multiple zeta value theory.

Findings

Explicit evaluations of signed and unsigned Tornheim double series

Introduction of signed q-analogs and their relation to double q-Euler sums

Corrections of previous errors in the literature

Abstract

We introduce signed q-analogs of Tornheim's double series, and evaluate them in terms of double q-Euler sums. As a consequence, we provide explicit evaluations of signed and unsigned Tornheim double series, and correct some mistakes in the literature.